In academic research paper all sections are linked:
Introduction ➡️ Literature Review ➡️ Method ➡️ Results ➡️
Discussion & Conclusion
To understand the statistics in the results section it is essential
to identify the concepts presented in each section:
Variables
A variable …
- Is way of assigning values (numbers or characters) to labels
- Corresponds to a column in a spreadsheet
Challenge: Identify the Role and the
Type of each variable
Type of
Variables
There are many type for variables but the only that we should care
are … - Continuous: If values are numbers -
Categorical: If values are characters
Note: Distinguish Categorical Nominal variables
(e.g., Irish, French) vs. Categorical Ordinal
variables (e.g., XS, S, M, L, XL)
Role of
Variables
A variable can have one or the other of these roles (no other role
exist):
- Outcome: “to be explained” variable as Y (also
called Dependent Variable or DV)
- Predictor: “doing the explaining” as X (also called
Independent Variable or IV)
Note: A variable can be also both but in different hypotheses
Hypotheses
Main Effect
Hypothesis Formulation
The Outcome has to be Continuous but …
- Case 1: Predictor is Continuous
The {outcome} increases when
{predictor} {increases/decreases/changes}
- Case 2: Predictor is Categorical (2 Categories)
The {outcome} of {predictor category
1} is {higher/lower/different} than the
{outcome} of {predictor category
2}
- Case 3: Predictor is Categorical (3 or more Categories)
The {outcome} of at least one
{predictor} category is
{higher/lower/different} than the other
{predictor} categories
Model &
Equation
The basic structure of a statistical model is:
\[Outcome = Model + Error\]
where the \(Model\) is a series of
predictors that are expressed in hypotheses related to the same
outcome:
- Main effect hypotheses are indicated with the predictor name
only
- Interaction effect hypotheses are indicated with all predictor names
separated by \(*\)
Example:
\[Outcome = Pred1 + Pred2 + Pred1 * Pred2
+ Error\]
To evaluate their relationship with the outcome, each effect
hypothesis is related with a coefficient called
Estimate and represented with \(\beta\) as follow:
\[Outcome = \beta_0 + \beta_1 Pred1 +
\beta_2 Pred2 + \beta_3 Pred1 * Pred2 + Error\]
Note: \(\beta_0\) is the estimate
related to the intercept. It is always included, always tested but has
no interest in the analysis
Evaluation of the
Significance
Testing for the significance of the effect means evaluating if this
estimate \(\beta\) value is
significantly different, higher or lower than 0 as
hypothesised in \(H_a\):
- \(\beta \neq 0\) means our
hypothesis doesn’t precise the direction of the change, just that there
is a change
- \(\beta > 0\) means our
hypothesis indicates that the relationship increases or a group is
higher than another group
- \(\beta < 0\) means our
hypothesis indicates that the relationship decreases or a group is lower
than another group
Note: \(H_0\) will always predict
that \(\beta = 0\)
The significance, called \(p\)-value, is the probability to consider
\(H_0\) as True. This probability is
between 0% and 100% which corresponds to a value between 0.0 and
1.0.
If the \(p\)-value:
- Is higher than 5% or 0.05, then \(H_0\) is accepted
- Is lower than 5% or 0.05, then \(H_0\) is rejected and
\(H_a\) is considered as plausible
Graphic
Representation of a Model
A graphic representation of the model’s hypothesised effects can be
done: - All the arrows correspond to an hypothesis to be tested - All
the tested hypotheses have to be represented with an arrow
A simple arrow is a main effect
A crossing arrow is an interaction effect
Note: By default, an interaction effect involves the test of the main
effect hypotheses of all Predictors involved
Statistical Test
JAMOVI: Stats. Open.
Now.
Jamovi an be downloaded or used online on https://www.jamovi.org/
A free book “Learning Statistics with Jamovi” by Navarro and Foxcroft
(2019) is available online here: https://www.learnstatswithjamovi.com/
Advantages:
- Free
- Simple Interface
- No Missing Values to Declare
- No Variable to Recode by Default
- Ready to Publish Tables and Figures
- Free Modules for Advanced Statistics (Mediation, Generalized LM,
Linear Mixed Model)
Note: In Jamovi …
- The outcome is called Dependent Variable
- A continuous predictor is a covariate
- A categorical predictor is a factor
Hypotheses with
Continuous Predictors and with Categorical Predictors Having 2
Categories
Steps:
- Open your file
- Check the type of your variables
- Analyses > Regression >
Linear Regression
- Set the Outcome as DV and
- To test the main effect hypotheses: set the
Predictors as Covariates/Factors
- To test interaction effect hypotheses: In Model
Builder, select all predictor with
CTRL (win) or
Command (mac) and bring them as interaction in the
model
Communicate the Results about the full model and each hypothesis:
- Use Model Fit Measure Table to evaluate the
accuracy of the full model
The predictions from a model including all effects are
significant/not-significant better than without these effects ( \(R^2 = value_{R^2}\), \(F(df1,df2) = value_{F}\), \(p = value_{p}\))
- Use Model Coefficients Table to conclude about each
hypothesis
The effect of \(Predictor\) on \(Outcome\) is statistically
significant/not-significant, therefore \(H_0\) can be rejected/accepted ( \(b = value_{estimate}, 95\% CI [lower\,CI,
upper\,CI]\), \(t(df) =
value_t\), \(p =
value_{p}\)).
Hypotheses with
Categorical Predictors Having 3 or more Categories
- Open your file
- Check the type of your variables
- Analyses > Regression >
Linear Regression
- Set the Outcome as DV and
- To test the main effect hypotheses: set the
Predictors as Factors
- To test interaction effect hypotheses: In Model
Builder options, select all predictor with
CTRL (win) or
Command (mac) and bring them as interaction in the
model
- Tick ANOVA Test in Model Coefficient options
Communicate the Results about the full model and each hypothesis:
- Use Model Fit Measure Table to evaluate the
accuracy of the full model
The predictions from a model including all effects are
significant/not-significant better than without these effects ( \(R^2 = value_{R^2}\), \(F(df1,df2) = value_{F}\), \(p = value_{p}\))
- Use Omnibus ANOVA Test Table to conclude about each
hypothesis
The effect of \(Predictor\) on \(Outcome\) is statistically
significant/not-significant, therefore \(H_0\) can be rejected/accepted ( \(F(df_{predictor}, df_{residual}) =
value_F\), \(p =
value_{p}\)).
Discussion &
Conclusion
From here…
- There is no number to be shown and no specific guidelines
- Correct interpretation comes if results have been understood and if
reasons for the results to be the ones obtained have been
identified
